Fluctuation-Dissipation Theorem in Nonequilibrium Steady States

Abstract

In equilibrium, the fluctuation-dissipation theorem (FDT) expresses the response of an observable to a small perturbation by a correlation function of this variable with another one that is conjugate to the perturbation with respect to energy. For a nonequilibrium steady state (NESS), the corresponding FDT is shown to involve in the correlation function a variable that is conjugate with respect to entropy. By splitting up entropy production into one of the system and one of the medium, it is shown that for systems with a genuine equilibrium state the FDT of the NESS differs from its equilibrium form by an additive term involving total entropy production. A related variant of the FDT not requiring explicit knowledge of the stationary state is particularly useful for coupled Langevin systems. The a priori surprising freedom apparently involved in different forms of the FDT in a NESS is clarified. Introduction. – Stochastic thermodynamics provides a framework for describing small driven systems embedded in a heat bath of still well-defined temperature [1]. Its crucial ingredients are a formulation of the first law [2] and the notion of a stochastic entropy [3] both valid along single fluctuating trajectories. Using these concepts several exact relations for distribution functions for quantities like work [4, 5] and entropy production [3, 6–9] have been derived. Experimental tests have been performed on a variety of different systems. Prominent examples include colloidal particles manipulated by laser traps [10–12], biomolecules pulled by AFM’s or optical tweezers [13, 14], and single defects observed using fluorescence techniques [15]. Reviews of this very active field can be found in Refs. [1, 16,17]. A particularly interesting class of states are nonequilibrium steady states (NESS) characterized both by a timeindependent distribution and, as a result of the external driving, nonvanishing currents. If such a NESS is perturbed by an additional small external force or field, one can ask whether the response of an observable of this system can be expressed by a correlation function involving this observable and a second one. For slightly perturbed equilibrium systems, such a connection between response and equilibrium fluctuation is given by the well-known fluctuation-dissipation theorem (FDT) [18,19]. The appropriate correlation function involves the observable whose response is sought for and another variable that is conjugate to the perturbation with respect to energy. The first purpose of this letter is to show that previously derived somewhat formal looking FDTs for general Markovian processes [20, 21] acquire a particularly simple and transparent form using the concepts of stochastic thermodynamics: In a nonequilibrium steady state, the response of a system to an additional small perturbation is given by a correlation function of this observable and another one that is conjugate to the perturbation with respect to stochastic entropy. Moreover, by expressing entropy production in the system as the difference between total entropy production and that in the surrounding medium, we can show that for a large class of systems the FDT in a NESS can be obtained from the corresponding equilibrium form of the FDT by subtracting a term involving total entropy production. The latter result rationalizes and generalizes recent results for diffusive systems driven by an external force [22, 23] or shear flow [24], see Refs. [25, 26] for first experimental tests of such extended FDTs. Adapting a recently introduced alternative strategy for deriving an FDT [27], we discuss a variant not requiring explicit knowledge of the typically unknown stationary distribution. This form will be particularly useful in simulations of coupled Langevin systems. Finally, we clarify the a priori surprising apparent freedom involved in different forms of the FDT in a NESS. For a broader overview of the FDT especially in systems with glassy dynamics, we refer to the review [28]. p-1 ar X iv :0 90 7. 54 78 v2 [ co nd -m at .s ta tm ec h] 1 5 D ec 2 00 9 Udo Seifert1 Thomas Speck2,3 Derivation of the FDT. – For a derivation of these results in a fairly general setting, we consider an arbitrary set of states {n}. These states could inter alia signify discrete spatial variables for a set of driven interacting diffusive degrees of freedom obtained by spatially discretizing Langevin equations. Likewise, they could code the states of any (bio)chemical reaction network. A transition from state m to n happens with a rate wmn(h), which depends on an external parameter h. The probability ψm(t) for finding the system in state m at time t obeys the master equation ∂tψm(t) = ∑

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Cite this paper

@inproceedings{Seifert2009FluctuationDissipationTI, title={Fluctuation-Dissipation Theorem in Nonequilibrium Steady States}, author={U. Seifert and Thomas Speck}, year={2009} }